Properties
Let A be a square n by n matrix over a field K (for example the field R of real numbers). The following statements are equivalent:
- A is invertible.
- A is row-equivalent to the n-by-n identity matrix In.
- A is column-equivalent to the n-by-n identity matrix In.
- A has n pivot positions.
- det A ≠ 0. In general, a square matrix over a commutative ring is invertible if and only if its determinant is a unit in that ring.
- rank A = n.
- The equation Ax = 0 has only the trivial solution x = 0 (i.e., Null A = {0})
- The equation Ax = b has exactly one solution for each b in Kn.
- The columns of A are linearly independent.
- The columns of A span Kn (i.e. Col A = Kn).
- The columns of A form a basis of Kn.
- The linear transformation mapping x to Ax is a bijection from Kn to Kn.
- There is an n by n matrix B such that AB = In = BA.
- The transpose AT is an invertible matrix (hence rows of A are linearly independent, span Kn, and form a basis of Kn).
- The number 0 is not an eigenvalue of A.
- The matrix A can be expressed as a finite product of elementary matrices.
Furthermore, the following properties hold for an invertible matrix A:
- (A−1)−1 = A;
- (kA)−1 = k−1A−1 for nonzero scalar k;
- (AT)−1 = (A−1)T;
- For any invertible n-by-n matrices A and B, (AB)−1 = B−1A−1. More generally, if A1,...,Ak are invertible n-by-n matrices, then (A1A2⋯Ak−1Ak)−1 = Ak−1Ak−1−1⋯A2−1A1−1;
- det(A−1) = det(A)−1.
A matrix that is its own inverse, i.e. A = A-1 and A2 = I, is called an involution.
Read more about this topic: Invertible Matrix
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